Theorems · Definition · ring theory

LinearMap.extendScalarsOfSurjectiveEquiv

{R : Type u_1} →
  {S : Type u_2} →
    [inst : CommSemiring R] →
      [inst_1 : Semiring S] →
        [inst_2 : Algebra R S] →
          {M : Type u_3} →
            {N : Type u_4} →
              [inst_3 : AddCommMonoid M] →
                [inst_4 : AddCommMonoid N] →
                  [inst_5 : Module R M] →
                    [inst_6 : Module S M] →
                      [IsScalarTower R S M] →
                        [inst_8 : Module R N] →
                          [inst_9 : Module S N] →
                            [inst_10 : IsScalarTower R S N] →
                              Function.Surjective ⇑(algebraMap R S) → (M →ₗ[R] N) ≃ₗ[R] M →ₗ[S] N

If R →+* S is surjective, then S-linear maps between modules are exactly R-linear maps.

Defined in

Mathlib.Algebra.Algebra.Basic

Cited by

2 results in Mathlib

Foundations

Depth 34 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringSemiringAlgebraAddCommMonoidAddCommMonoidModuleModuleIsScalarTowerModuleModuleIsScalarTower